special elements in a lattice
Let $L$ be a lattice^{} and $a\in L$ is said to be

•
distributive if $a\vee (b\wedge c)=(a\vee b)\wedge (a\vee c)$,

•
standard if $b\wedge (a\vee c)=(b\wedge a)\vee (b\wedge c)$, or

•
neutral if $(a\wedge b)\vee (b\wedge c)\vee (c\wedge a)=(a\vee b)\wedge (b\vee c)\wedge (c\vee a)$
for all $b,c\in L$. There are also dual notions of the three types mentioned above, simply by exchanging $\vee $ and $\wedge $ in the definitions. So a dually distributive element $a\in L$ is one where $a\wedge (b\vee c)=(a\wedge b)\vee (a\wedge c)$ for all $b,c\in L$, and a dually standard element is similarly defined. However, a dually neutral element is the same as a neutral element.
Remarks For any $a\in L$, suppose $P$ is the property in $L$ such that $a\in P$ iff $a\vee b=a\vee c$ and $a\wedge b=a\wedge c$ imply $b=c$ for all $b,c\in L$.

•
A standard element is distributive. Conversely, a distributive satisfying $P$ is standard.

•
A neutral element is distributive (and consequently dually distributive). Conversely, a distributive and dually distributive element that satisfies $P$ is neutral.
References
 1 G. Birkhoff Lattice Theory, 3rd Edition, AMS Volume XXV, (1967).
 2 G. Grätzer, General Lattice Theory, 2nd Edition, Birkhäuser (1998).
Title  special elements in a lattice 

Canonical name  SpecialElementsInALattice 
Date of creation  20130322 16:42:29 
Last modified on  20130322 16:42:29 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  6 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 06B99 
Defines  distributive element 
Defines  standard element 
Defines  neutral element 
Defines  dually distributive 
Defines  dually standard 